#419462

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,12,17,11)(8,19)(9,22)(10,14)(13,20)(15,25)(16,32,33,36)(18,27,23,29)(21,31)(24,30)(26,34,28,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,23)(10,17)(12,14)(13,28)(15,29)(18,25)(19,36)(20,35)(21,33)(22,27)(24,34)(26,30)(31,32);(2,4)(5,7)(6,12,14,11)(8,16)(9,18)(10,17)(13,28,30,26)(15,29,22,27)(19,32,31,36)(20,34)(21,33)(23,25)(24,35);(5,8,10,21)(6,13,14,30)(7,16,17,33)(9,24,25,20)(11,26,12,28)(15,31,22,19)(18,35,23,34)(27,32,29,36)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326658);